Weak uniqueness by noise for singular stochastic PDEs

TL;DR

This paper establishes weak uniqueness for a broad class of stochastic PDEs with irregular drifts and various noise types, expanding the understanding of solution uniqueness without requiring H"older continuity.

Contribution

It introduces a novel framework for proving weak uniqueness of SPDEs with drifts defined on Sobolev spaces, applicable to diverse noise structures.

Findings

Proves weak uniqueness for SPDEs with non-H"older continuous drifts.

Extends results to SPDEs driven by coloured and rougher noises.

Provides a unified approach for multiple specific examples.

Abstract

We prove weak uniqueness of mild solutions for general classes of SPDEs on a Hilbert space. The main novelty is that the drift is only defined on a Sobolev-type subspace and no H\"older-continuity assumptions are required. This framework turns out to be effective to achieve novel uniqueness results for several specific examples. Such wide range of applications is obtained by exploiting either coloured or rougher-than-cylindrical noises.